+Intuitively, (a) and (b) express the application of the same function to the argument `y`:

+

+<OL type=a>

+<LI><code>(\x. \z. z x) y</code>

+<LI><code>(\x. \y. y x) y</code>

+</OL>

+

+One can't just rename variables freely. (a) and (b) are different than what's expressed by:

+

+<OL type=a start=3>

+<LI><code>(\z. (\z. z z) y</code>

+</OL>

+

+

+Substituting `y` into the body of (a) `(\x. \z. z x)` is unproblematic:

+

+ (\x. \z. z x) y ~~> \z. z y

+

+However, with (b) we have to be more careful. If we just substituted blindly, then we might take the result to be `\y. y y`. But this is the self-application function, not the function which accepts an arbitrary argument and applies that argument to the free variable `y`. In fact, the self-application function is what (c) reduces to. So if we took (b) to reduce to `\y. y y`, we'd wrongly be counting (b) to be equivalent to (c), instead of (a).

+

+To reduce (b), then, we need to be careful to that no free variables in what we're substituting in get captured by binding &lambda;s that they shouldn't be captured by.

+

+In practical terms, you'd just replace (b) with (a) and do the unproblematic substitution into (a).

+

+How should we think about the explanation and justification for that practical procedure?

+

+One way to think about things here is to identify expressions of the lambda calculus with *particular alphabetic sequences*. Then (a) and (b) would be distinct expressions, and we'd have to have an explicit rule permitting us to do the kind of variable-renaming that takes us from (a) to (b) (or vice versa). This kind of renaming is called "alpha-conversion." Look in the standard treatments of the lambda calculus for detailed discussion of this.

+

+Another way to think of it is to identify expressions not with particular alphabetic sequences, but rather with *classes* of alphabetic sequences, which stand to each other in the way that (a) and (b) do. That's the way we'll talk. We say that (a) and (b) are just typographically different notations for a *single* lambda formula. As we'll say, the lambda formula written with (a) and the lambda formula written with (b) are literally syntactically identical.

+

+A third way to think is to identify the lambda formula not with classes of alphabetic sequences, but rather with abstract structures that we might draw like this:

+

+<pre><code>

+ (&lambda;. &lambda;. _ _) y

+ ^ ^ | |

+ | |__| |

+ |_______|

+</code></pre>

+

+Here there are no bound variables, but there are *bound positions*. We can regard formula like (a) and (b) as just helpfully readable ways to designate these abstract structures.

+

+A version of this last approach is known as **de Bruijn notation** for the lambda calculus.

+

+It doesn't seem to matter which of these approaches one takes; the logical properties of the systems are exactly the same. It just affects the particulars of how one states the rules for substitution, and so on. And whether one talks about expressions being literally "syntactically identical," or whether one instead counts them as "equivalent modulu alpha-conversion."

+

+(Linguistic trivia: however, some linguistic discussions do suppose that alphabetic variance has important linguistic consequences; see Ivan Sag's dissertation.)

+

+In a bit, we'll discuss other systems that lack variables. Those systems will not just lack variables in the sense that de Bruijn notation does; they will furthermore lack any notion of a bound position.

+

+

Syntactic equality, reduction, convertibility

=============================================

-Define T to be `(\x. x y) z`. Then T and `(\x. x y) z` are syntactically equal, and we're counting them as syntactically equal to `(\z. z y) z` as well, which we will write as:

+Define N to be `(\x. x y) z`. Then N and `(\x. x y) z` are syntactically equal, and we're counting them as syntactically equal to `(\z. z y) z` as well, which we will write as:

-<pre><code>T &equiv; (\x. x y) z &equiv; (\z. z y) z

+<pre><code>N &equiv; (\x. x y) z &equiv; (\z. z y) z

</code></pre>

This:

- T ~~> z y

+ N ~~> z y

-means that T beta-reduces to `z y`. This:

+means that N beta-reduces to `z y`. This:

- M <~~> T

+ M <~~> N

-means that M and T are beta-convertible, that is, that there's something they both reduce to in zero or more steps.

+means that M and N are beta-convertible, that is, that there's something they both reduce to in zero or more steps.

Combinators and Combinatorial Logic

===================================

@@ -23,21+78,181 @@ Lambda expressions that have no free variables are known as **combinators**. Her

> **I** is defined to be `\x x`

-> **K** is defined to be `\x y. x`, That is, it throws away its second argument. So `K x` is a constant function from any (further) argument to `x`. ("K" for "constant".) Compare K to our definition of **true**.

+> **K** is defined to be `\x y. x`. That is, it throws away its

+ second argument. So `K x` is a constant function from any

+ (further) argument to `x`. ("K" for "constant".) Compare K

+ to our definition of `true`.

+

+> **get-first** was our function for extracting the first element of an ordered pair: `\fst snd. fst`. Compare this to K and `true` as well.

+

+> **get-second** was our function for extracting the second element of an ordered pair: `\fst snd. snd`. Compare this to our definition of `false`.

-> **get-first** was our function for extracting the first element of an ordered pair: `\fst snd. fst`. Compare this to **K** and **true** as well.

It's possible to build a logical system equally powerful as the lambda calculus (and readily intertranslatable with it) using just combinators, considered as atomic operations. Such a language doesn't have any variables in it: not just no free variables, but no variables at all.

One can do that with a very spare set of basic combinators. These days the standard base is just three combinators: K and I from above, and also one more, **S**, which behaves the same as the lambda expression `\f g x. f x (g x)`. behaves. But it's possible to be even more minimalistic, and get by with only a single combinator. (And there are different single-combinator bases you can choose.)

-These systems are Turing complete. In other words: every computation we know how to describe can be represented in a logical system consisting of only a single primitive operation!

+There are some well-known linguistic applications of Combinatory

+Logic, due to Anna Szabolcsi, Mark Steedman, and Pauline Jacobson.

+They claim that natural language semantics is a combinatory system: that every

+natural language denotation is a combinator.

+

+For instance, Szabolcsi argues that reflexive pronouns are argument

+duplicators.

+

+![reflexive](http://lambda.jimpryor.net/szabolcsi-reflexive.jpg)

+

+Notice that the semantic value of *himself* is exactly `W`.

+The reflexive pronoun in direct object position combines with the transitive verb. The result is an intransitive verb phrase that takes a subject argument, duplicates that argument, and feeds the two copies to the transitive verb meaning.

+we can define combinators by what they do. If we have the I combinator followed by any expression X,

+I will take that expression as its argument and return that same expression as the result. In pictures,

+

+ IX ~~> X

+

+Thinking of this as a reduction rule, we can perform the following computation

+

+ II(IX) ~~> IIX ~~> IX ~~> X

+

+The reduction rule for K is also straightforward:

+

+ KXY ~~> X

+

+That is, K throws away its second argument. The reduction rule for S can be constructed by examining

+the defining lambda term:

+

+<pre><code>S &equiv; \fgx.fx(gx)</code></pre>

+

+S takes three arguments, duplicates the third argument, and feeds one copy to the first argument and the second copy to the second argument. So:

-Here's more to read about combinatorial logic:

+ SFGX ~~> FX(GX)

+

+If the meaning of a function is nothing more than how it behaves with respect to its arguments,

+these reduction rules capture the behavior of the combinators S, K, and I completely.

+We can use these rules to compute without resorting to beta reduction. For instance, we can show how the I combinator is equivalent to a certain crafty combination of Ss and Ks:

+

+ SKKX ~~> KX(KX) ~~> X

+

+So the combinator `SKK` is equivalent to the combinator I.

+

+Combinatory Logic is what you have when you choose a set of combinators and regulate their behavior with a set of reduction rules. As we said, the most common system uses S, K, and I as defined here.

+

+###The equivalence of the untyped lambda calculus and combinatory logic###

+

+We've claimed that Combinatory Logic is equivalent to the lambda calculus. If that's so, then S, K, and I must be enough to accomplish any computational task imaginable. Actually, S and K must suffice, since we've just seen that we can simulate I using only S and K. In order to get an intuition about what it takes to be Turing complete, imagine what a text editor does:

+it transforms any arbitrary text into any other arbitrary text. The way it does this is by deleting, copying, and reordering characters. We've already seen that K deletes its second argument, so we have deletion covered. S duplicates and reorders, so we have some reason to hope that S and K are enough to define arbitrary functions.

+

+We've already established that the behavior of combinatory terms can be perfectly mimicked by lambda terms: just replace each combinator with its equivalent lambda term, i.e., replace I with `\x.x`, replace K with `\fxy.x`, and replace S with `\fgx.fx(gx)`. How about the other direction? Here is a method for converting an arbitrary lambda term into an equivalent Combinatory Logic term using only S, K, and I. Besides the intrinsic beauty of this mapping, and the importance of what it says about the nature of binding and computation, it is possible to hear an echo of computing with continuations in this conversion strategy (though you wouldn't be able to hear these echos until we've covered a considerable portion of the rest of the course).

+

+Assume that for any lambda term T, [T] is the equivalent combinatory logic term. The we can define the [.] mapping as follows:

+

+ 1. [a] a

+ 2. [(M N)] ([M][N])

+ 3. [\a.a] I

+ 4. [\a.M] KM assumption: a does not occur free in M

+ 5. [\a.(M N)] S[\a.M][\a.N]

+ 6. [\a\b.M] [\a[\b.M]]

+

+It's easy to understand these rules based on what S, K and I do. The first rule says

+that variables are mapped to themselves.

+The second rule says that the way to translate an application is to translate the

+first element and the second element separately.

+The third rule should be obvious.

+The fourth rule should also be fairly self-evident: since what a lambda term such as `\x.y` does it throw away its first argument and return `y`, that's exactly what the combinatory logic translation should do. And indeed, `Ky` is a function that throws away its argument and returns `y`.

+The fifth rule deals with an abstract whose body is an application: the S combinator takes its next argument (which will fill the role of the original variable a) and copies it, feeding one copy to the translation of \a.M, and the other copy to the translation of \a.N. This ensures that any free occurrences of a inside M or N will end up taking on the appropriate value. Finally, the last rule says that if the body of an abstract is itself an abstract, translate the inner abstract first, and then do the outermost. (Since the translation of [\b.M] will not have any lambdas in it, we can be sure that we won't end up applying rule 6 again in an infinite loop.)

+

+[Fussy notes: if the original lambda term has free variables in it, so will the combinatory logic translation. Feel free to worry about this, though you should be confident that it makes sense. You should also convince yourself that if the original lambda term contains no free variables---i.e., is a combinator---then the translation will consist only of S, K, and I (plus parentheses). One other detail: this translation algorithm builds expressions that combine lambdas with combinators. For instance, the translation of our boolean false `\x.\y.y` is `[\x[\y.y]] = [\x.I] = KI`. In the intermediate stage, we have `\x.I`, which mixes combinators in the body of a lambda abstract. It's possible to avoid this if you want to, but it takes some careful thought. See, e.g., Barendregt 1984, page 156.]

+

+[Various, slightly differing translation schemes from combinatorial logic to the lambda calculus are also possible. These generate different metatheoretical correspondences between the two calculii. Consult Hindley and Seldin for details. Also, note that the combinatorial proof theory needs to be strengthened with axioms beyond anything we've here described in order to make [M] convertible with [N] whenever the original lambda-terms M and N are convertible.]

+

+

+Let's check that the translation of the false boolean behaves as expected by feeding it two arbitrary arguments:

+

+ KIXY ~~> IY ~~> Y

+

+Throws away the first argument, returns the second argument---yep, it works.

+

+Here's a more elaborate example of the translation. The goal is to establish that combinators can reverse order, so we use the **T** combinator, where <code>T &equiv; \x\y.yx</code>:

In the assignment we asked you to reduce various expressions until it wasn't possible to reduce them any further. For two of those expressions, this was impossible to do. One of them was this:

@@ -97,16+312,33 @@ This question highlights that there are different choices to make about how eval

With regard to Q3, it should be intuitively clear that `\x. M x` and `M` will behave the same with respect to any arguments they are given. It can also be proven that no other functions can behave differently with respect to them. However, the logical system you get when eta-reduction is added to the proof theory is importantly different from the one where only beta-reduction is permitted.

-MORE on extensionality

-

-If we answer Q2 by permitting reduction inside abstracts, and we also permit eta-reduction, then where neither `y` nor `z` occur in M, this:

+If we answer Q2 by permitting reduction inside abstracts, and we also permit eta-reduction, then where none of <code>y<sub>1</sub>, ..., y<sub>n</sub></code> occur free in M, this:

+When we add eta-reduction to our proof system, we end up reconstruing the meaning of `~~>` and `<~~>` and "normal form", all in terms that permit eta-reduction as well. Sometimes these expressions will be annotated to indicate whether only beta-reduction is allowed (<code>~~><sub>&beta;</sub></code>) or whether both beta- and eta-reduction is allowed (<code>~~><sub>&beta;&eta;</sub></code>).

+

+The logical system you get when eta-reduction is added to the proof system has the following property:

+

+> if `M`, `N` are normal forms with no free variables, then <code>M &equiv; N</code> iff `M` and `N` behave the same with respect to every possible sequence of arguments.

+

+This implies that, when `M` and `N` are (closed normal forms that are) syntactically distinct, there will always be some sequences of arguments <code>L<sub>1</sub>, ..., L<sub>n</sub></code> such that:

+

+<pre><code>M L<sub>1</sub> ... L<sub>n</sub> x y ~~> x

+N L<sub>1</sub> ... L<sub>n</sub> x y ~~> y

+</code></pre>

+

+So closed beta-plus-eta-normal forms will be syntactically different iff they yield different values for some arguments. That is, iff their extensions differ.

+

+So the proof theory with eta-reduction added is called "extensional," because its notion of normal form makes syntactic identity of closed normal forms coincide with extensional equivalence.

+

+See Hindley and Seldin, Chapters 7-8 and 14, for discussion of what should count as capturing the "extensionality" of these systems, and some outstanding issues.

+

+

The evaluation strategy which answers Q1 by saying "reduce arguments first" is known as **call-by-value**. The evaluation strategy which answers Q1 by saying "substitute arguments in unreduced" is known as **call-by-name** or **call-by-need** (the difference between these has to do with efficiency, not semantics).

When one has a call-by-value strategy that also permits reduction to continue inside unapplied abstracts, that's known as "applicative order" reduction. When one has a call-by-name strategy that permits reduction inside abstracts, that's known as "normal order" reduction. Consider an expression of the form:

@@ -161,7+393,8 @@ One important advantage of normal-order evaluation in particular is that it can

Indeed, it's provable that if there's *any* reduction path that delivers a value for a given expression, the normal-order evalutation strategy will terminate with that value.

-An expression is said to be in **normal form** when it's not possible to perform any more reductions. (EVEN INSIDE ABSTRACTS?) There's a sense in which you *can't get anything more out of* <code>&omega; &omega;</code>, but it's not in normal form because it still has the form of a redex.

+An expression is said to be in **normal form** when it's not possible to perform any more reductions (not even inside abstracts).

+There's a sense in which you *can't get anything more out of* <code>&omega; &omega;</code>, but it's not in normal form because it still has the form of a redex.

A computational system is said to be **confluent**, or to have the **Church-Rosser** or **diamond** property, if, whenever there are multiple possible evaluation paths, those that terminate always terminate in the same value. In such a system, the choice of which sub-expressions to evaluate first will only matter if some of them but not others might lead down a non-terminating path.

@@ -203,6+436,7 @@ But is there any method for doing this in general---for telling, of any given co

* [Scooping the Loop Snooper](http://www.cl.cam.ac.uk/teaching/0910/CompTheory/scooping.pdf), a proof of the undecidability of the halting problem in the style of Dr Seuss by Geoffrey K. Pullum

+Interestingly, Church also set up an association between the lambda calculus and first-order predicate logic, such that, for arbitrary lambda formulas `M` and `N`, some formula would be provable in predicate logic iff `M` and `N` were convertible. So since the right-hand side is not decidable, questions of provability in first-order predicate logic must not be decidable either. This was the first proof of the undecidability of first-order predicate logic.